Parametric and Nonparametric Bayesian Models for Ecological Inference in 2 ◊ 2 Tables
Kosuke Imai, Ying Lü · 2004
The ecological inference problem arises when making inferences about individual behavior from aggregate data. Such a situation is frequently encountered in the social sciences and epidemiology. In this article, we propose a Bayesian approach based on data augmentation. We formulate ecological inference in 2◊2 tables as a missing data problem where only the weighted average of two unknown variables is observed. This framework directly incorporates the deterministic bounds, which contain all information available from the data, and allow researchers to incorporate the individual-level data whenever available. Within this general framework, we first develop a parametric model. We show that through the use of an EM algorithm, the model can formally quantify the eect of missing information on parameter estimation. This is an important diagnostic for evaluating the degree of aggregation eects. Next, we introduce a nonparametric model using a Dirichlet process prior to relax the distributional assumption of the parametric model. Through simulations and an empirical application, we evaluate the relative performance of our models in various situations. We demonstrate that in typical ecological inference problems, the fraction of missing information often exceeds 50 percent. We also find that the nonparametric model generally outperforms the parametric model, although the latter gives reasonable in-sample predictions when the bounds are informative. C-code, along with an R interface, is publicly available for implementing our Markov chain Monte Carlo algorithms to